Question

The value closest to the thermal velocity of a Helium atom at room temperature $(300\, K)$ in $ms^{-1}$ is :$ [k_B = 1.4 \times 10^{-23} \, J/K; m_{He} = 7 \times 10^{-27} \, kg ]$

• A. $1.3 \times 10^4$
• B. $1.3 \times 10^3$
• C. $1.3 \times 10^5$
• D. $1.3 \times 10^2$

Answer 1

The Correct Option is B

We know that $v_{ rms }=\sqrt{\frac{3 k T}{m}}$. Given, $k=1.4 \times 10^{-23} \,J / K ; T=300 \,K ; m=7 \times 10^{-27}\, kg$. Therefore, $v_{ rms } =\sqrt{\frac{3 \times 1.4 \times 10^{-23} \times 300}{7 \times 10^{-27}}}=\sqrt{\frac{3 \times 300 \times 14 \times 10^{-23}}{7 \times 10 \times 10^{-27}}}=\sqrt{\frac{3^{2} \times 10^{2} \times 2 \times 10^{4}}{10}} $ $=3 \times 10 \times 10^{2} \times \sqrt{\frac{2}{10}} $ $=3 \times \sqrt{10} \times 10^{2} \times \sqrt{2}=3 \times \sqrt{2} \times \sqrt{5} \times 10^{2} \times \sqrt{2}=3 \times 2 \times \sqrt{5} \times 10^{2}$ $v_{ rms } =6 \times 10^{2} \times \sqrt{5}=13.41 \times 10^{2}$ or $ v_{ rms }=1.34 \times 10^{3}$